Chapter Four,Utility,Structure,Utility function (效用函数） Definition Monotonic transformation （单调转换） Examples of utility functions and their indifference curves Marginal utility （边际效用） Marginal rate of substitution 边际替代率 MRS after monotonic transformation,Utility Functions,A utility function U(x) represents a preference relation if and only if: x x” U(x) U(x”) x x” U(x) U(x”) x x” U(x) = U(x”).,p,p,Utility Functions,Utility is an ordinal (i.e. ordering) concept. 序数效用 E.g. if U(x) = 6 and U(y) = 2 then bundle x is strictly preferred to bundle y. But x is not preferred three times as much as is y.,Utility Functions & Indiff. Curves,Consider the bundles (4,1), (2,3) and (2,2). Suppose (2,3) (4,1) (2,2). Assign to these bundles any numbers that preserve the preference ordering; e.g. U(2,3) = 6 U(4,1) = U(2,2) = 4. Call these numbers utility levels.,p,Utility Functions & Indiff. Curves,An indifference curve contains equally preferred bundles. Equal preference same utility level. Therefore, all bundles in an indifference curve have the same utility level.,Utility Functions & Indiff. Curves,So the bundles (4,1) and (2,2) are in the indiff. curve with utility level U 4 But the bundle (2,3) is in the indiff. curve with utility level U 6. On an indifference curve diagram, this preference information looks as follows:,Utility Functions & Indiff. Curves,U 6,U 4,(2,3) (2,2) (4,1),x1,x2,p,Utility Functions & Indiff. Curves,Comparing more bundles will create a larger collection of all indifference curves and a better description of the consumers preferences.,Utility Functions & Indiff. Curves,U 6,U 4,U 2,x1,x2,Utility Functions & Indiff. Curves,The collection of all indifference curves for a given preference relation is an indifference map. An indifference map is equivalent to a utility function; each is the other.,Utility Functions,There is no unique utility function representation of a preference relation. Suppose U(x1,x2) = x1x2 represents a preference relation. Again consider the bundles (4,1), (2,3) and (2,2).,Utility Functions,U(x1,x2) = x1x2, so U(2,3) = 6 U(4,1) = U(2,2) = 4; that is, (2,3) (4,1) (2,2).,p,Utility Functions,U(x1,x2) = x1x2 (2,3) (4,1) (2,2). Define V = U2.,p,Utility Functions,U(x1,x2) = x1x2 (2,3) (4,1) (2,2). Define V = U2. Then V(x1,x2) = x12x22 and V(2,3) = 36 V(4,1) = V(2,2) = 16 so again (2,3) (4,1) (2,2). V preserves the same order as U and so represents the same preferences.,p,p,Utility Functions,U(x1,x2) = x1x2 (2,3) (4,1) (2,2). Define W = 2U + 10.,p,Utility Functions,U(x1,x2) = x1x2 (2,3) (4,1) (2,2). Define W = 2U + 10. Then W(x1,x2) = 2x1x2+10 so W(2,3) = 22 W(4,1) = W(2,2) = 18. Again, (2,3) (4,1) (2,2). W preserves the same order as U and V and so represents the same preferences.,p,p,Utility Functions: Monotonic Transformation,If U is a utility function that represents a preference relation and f is a strictly increasing function, then V = f(U) is also a utility function representing .,Goods, Bads and Neutrals,A good is a commodity unit which increases utility (gives a more preferred bundle). A bad is a commodity unit which decreases utility (gives a less preferred bundle). A neutral is a commodity unit which does not change utility (gives an equally preferred bundle).,Goods, Bads and Neutrals,Utility,Water,x,Units of water are goods,Units of water are bads,Around x units, a little extra water is a neutral.,Utility function,Some Other Utility Functions and Their Indifference Curves,Instead of U(x1,x2) = x1x2 consider V(x1,x2) = x1 + x2. What do the indifference curves for this “perfect substitution” utility function look like?,Perfect Substitution Indifference Curves,5,5,9,9,13,13,x1,x2,x1 + x2 = 5,x1 + x2 = 9,x1 + x2 = 13,V(x1,x2) = x1 + x2.,Perfect Substitution Indifference Curves,5,5,9,9,13,13,x1,x2,x1 + x2 = 5,x1 + x2 = 9,x1 + x2 = 13,All are linear and parallel.,V(x1,x2) = x1 + x2.,Some Other Utility Functions and Their Indifference Curves,Instead of U(x1,x2) = x1x2 or V(x1,x2) = x1 + x2, consider W(x1,x2) = minx1,x2. What do the indifference curves for this “perfect complementarity” utility function look like?,Perfect Complementarity Indifference Curves,x2,x1,45o,minx1,x2 = 8,3,5,8,3,5,8,minx1,x2 = 5,minx1,x2 = 3,W(x1,x2) = minx1,x2,Perfect Complementarity Indifference Curves,x2,x1,45o,minx1,x2 = 8,3,5,8,3,5,8,minx1,x2 = 5,minx1,x2 = 3,All are right-angled with vertices on a ray from the origin.,W(x1,x2) = minx1,x2,Some Other Utility Functions and Their Indifference Curves,A utility function of the form U(x1,x2) = f(x1) + x2 is linear in just x2 and is called quasi-linear （准线性）. E.g. U(x1,x2) = 2x11/2 + x2.,Quasi-linear Indifference Curves,x2,x1,Each curve is a vertically shifted copy of the others.,Some Other Utility Functions and Their Indifference Curves,Any u